An Overlapping Schwarz Algorithm for Raviart--Thomas Vector Fields with Discontinuous Coefficients

Abstract

Overlapping Schwarz methods form one of two major families of domain decomposition methods. In this paper, a two-level overlapping Schwarz method for Raviart--Thomas vector fields is developed and analyzed. The global component of the preconditioner is based on local energy-minimizing discrete harmonic extensions of coarse interface values, and the local components use local direct solvers on overlapping subdomains. The algorithm can be implemented algebraically using submatrices of a fully assembled stiffness matrix of the entire finite element problem. Moreover, it gives a condition number bound which depends only on the size of the local subproblems. The condition number of our method is shown to be bounded by $C ( 1 + \log (H/h)) ( 1 + H/\delta)$, where $H$ is the diameter of the subdomain and $h$ and $\delta$ are the sizes of the elements and the overlap between subdomains, respectively. The bound of the condition number is independent of the values and jumps of the coefficients across the interface between the subdomains. Numerical results, which support the theory, are also presented.